Beauty In Polar Coordinates

This post is a part of a series of guest-posts on polar coordinates and complex numbers. These posts were written by my pre-calc students:

Beauty in Polar Coordinates

by Luke VanDyke

Beauty can be found in mathematics in various places. Whether it be in Euler’s Identity or another beautiful equation of the sort, or in a magnificent graph, order and beauty are found on every page. One of the most interesting and amazing ways to graph objects is in polar form. A variety of shapes such as spirals, cardioids, and limaçon are just a few examples of the wide range of beauty found in graphing in the polar form. I think one of the most amazing curves that you can graph is the rose. Using Geogebra, I was able to explore in great deal the immense complexity of such an amazing curve and also notice several key patterns regarding the equation.



In order to fully demonstrate the beauty of the rose, I first inserted two sliders, a and b. I made the range for each slider from 0-10 with increments of .1. These sliders would serve as my values for a and n. The equation we learned in class (r=a*cos(nθ) and r=a*sin(θ)) must be changed into the curve expression on Geogebra. Mr. Roer helped me out a lot in converting to curve form. Once I had the equation in, I was able to play with the sliders and see how they worked. Slider A adjusts the size of the radius. The larger the number on the slider, the larger the rose will “grow”.

Slider B was a lot of fun to play with. Slider B adjusted the number and also the width of the “petals”. After playing around with both sliders, I noticed a pattern developing that affected the graph significantly. When the value of slider B was odd(let’s call this value x), the flower would have x number of petals. For example, if Slider B=1, there was just one circle. The same occurred for all other odd values between 0-10. On the other hand, even numbers also had an interesting pattern. For every even value between 0-10(we’ll call this value y), there would be 2y number of petals. For example, if 2 was the value on the B slider, the rose would have 4 petals.


This project was a really great opportunity to see God’s beauty as seen in mathematics. The saying “Out of intense complexities, intense simplicities emerge.” really goes hand in hand with this project. Once you put in the big, complex equation, a simplistic, beautiful image of a rose emerges. This saying also applies to roses in real life. Out of a complex equation of photosynthesis and many other factors, a beautiful flower emerges. There are many other examples all throughout God’s magnificent creation. Whether it be in the majestic landscapes throughout nature or in the complexity of infinitesimal human DNA and genetics, God’s awesome handiwork is seen all throughout the earth. Through this project, I’ve been able to reflect on how great God is and how everything in His creation is simply remarkable, even in places where you least expect it.